Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 74
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Becauseof these advantages, the offset strip array has been studied by many researchers andused widely in compact heat exchangers.Figure 6.18 includes a table showing the relative strip thicknesses (t/) and therelative strip spacings (s/) studied by DeJong et al. (1998). Those rectangles paintedblack in the sketch of the strip array are the strips covered with naphthalene to measurethe mass transfer rate. The mass transfer data were converted to the heat transfercoefficient using the analogy between mass and heat transfer. The experimental datareveal row-by-row variations of heat transfer coefficient that depend on the Reynoldsnumber.
Figure 6.19 shows the friction factor f and Colburn j -factor j plotted as afunction of the Reynolds number Re. They are defined asRow1 2 3 4 5 6 7 8sFlowltGeometrical parameters for experimental and numerical arraysgeometricalexperimentalnumericalparametergeometrygeometryt /l0.1250.117s/l0.3750.507Figure 6.18 Offset strips. Black strips were naphthalene coated to measure the mass transfercoefficient. The table shows the dimensions. (From DeJong et al., 1998.)BOOKCOMP, Inc. — John Wiley & Sons / Page 487 / 2nd Proofs / Heat Transfer Handbook / Bejan[487], (49)Lines: 2317 to 2334———0.927pt PgVar———Normal PagePgEnds: TEX[487], (49)4881numerical simulationJoshi and Webb (1987) correlation0.1j123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWS0.01[488], (50)Lines: 2334 to 23600.00110010001,000ReFigure 6.19 Colburn j -factors and friction factors versus Reynolds number.
The curves werebased on the Joshi and Webb correlations [eqs. (6.166)–(6.169)]. (From DeJong et al., 1998.)Re =Uc dhν2(s − t)l+tν is the fluid kinematic viscosity, andf =2∆p core dhρUc2 4Lcore(6.163)where ∆pcore is the pressure drop across the entire eight-row test section, Lcore is thetotal length of the strip array, ρ is the fluid density, andj=Nu mPrRe(6.164)where Nu is the Nusselt number, and m = 0.40 [DeJong et al. (1998)] or 0.30 [Joshiand Webb (1987)].
The original j -factor in DeJong et al. (1998) is defined using theBOOKCOMP, Inc. — John Wiley & Sons / Page 488 / 2nd Proofs / Heat Transfer Handbook / Bejan———Normal PagePgEnds: TEX[488], (50)(6.162)where Uc is the flow velocity at the minimum free-flow area, dh is the hydraulicdiameter,dh =———0.34215pt PgVarHEAT TRANSFER FROM ARRAYS OF OBJECTS123456789101112131415161718192021222324252627282930313233343536373839404142434445489Sherwood and Schmidt numbers. For the sake of consistency throughout this section,they are replaced by heat transfer parameters indh havkNu =(6.165)where hav is the average heat transfer coefficient and k is the fluid thermal conductivity.The curves in Fig.
6.19 show the correlations proposed by Joshi and Webb (1987).With the transition Reynolds number denoted as Re*, in the laminar flow range whereRe ≤ Re∗ ,f = 8.12Re−0.74j = 0.53Re−0.50dhdh−0.41−0.15α−0.02(6.166)α−0.14(6.167)Lines: 2360 to 2413where α is the aspect ratio; α = s/W , with W taken as the strip width measurednormal to the page in Fig. 6.18. In the turbulent flow range where Re ≥ Re∗ + 1000,f = 1.12Re−0.36j = 0.21Re−0.40dhdh−0.65 −0.24 tdhtdh0.17(6.168)dhb(6.169)(6.170)withRe∗b 1.23 0.58t= 257s(6.171)1.328(Re )1/2(6.172)Uc ν(6.173)b=t+andRe =The solid symbols in Fig.
6.19 are the results of numerical simulation on the samestrip. Figure 6.19 shows the state-of-the-art accuracy in the predictions of f and j .BOOKCOMP, Inc. — John Wiley & Sons / Page 489 / 2nd Proofs / Heat Transfer Handbook / Bejan———5.00323pt PgVar———Normal PagePgEnds: TEX0.02The transition Reynolds number Re∗ is given byRe∗ = Re∗b[489], (51)[489], (51)4901234567891011121314151617181920212223242526272829303132333435363738394041424344456.6FORCED CONVECTION: EXTERNAL FLOWSHEAT TRANSFER FROM OBJECTS ON A SUBSTRATEFigure 6.20 shows a classification of the situations and models encountered in electronics cooling applications: (a) a heated strip that is flush bonded to the substratesurface, (b) a rectangular heat source which is also flush to the substrate surface, (c)an isolated two-dimensional block, (d) a two-dimensional block array, (e) a rectangular block and ( f ) an array of rectangular blocks.
Ortega et al. (1994) suggest thatin many practical situations the substrate is a heat conductor, so that a heat path fromthe heat source through the substrate to the fluid flow cannot be ignored in what iscalled conductive/convective conjugate heat transfer. The analytical solution of the[490], (52)Lines: 2413 to 2425———0.757pt PgVar———Short PagePgEnds: TEX[490], (52)Figure 6.20 Configurations of heat sources on substrate.
(From Ortega et al., 1994.)BOOKCOMP, Inc. — John Wiley & Sons / Page 490 / 2nd Proofs / Heat Transfer Handbook / BejanHEAT TRANSFER FROM OBJECTS ON A SUBSTRATE123456789101112131415161718192021222324252627282930313233343536373839404142434445491conjugate heat transfer problem is difficult, and numerical solutions or experimentshave been used in the studies reported in the literature.6.6.1Flush-Mounted Heat SourcesHeat transfer considerations in flush-mounted heat sources involve less complicationsthan in those for fluid flow.
The flow is described by the solution for boundary layerflow or duct flow, particularly where the flow is laminar and conjugate heat transferis important in laminar flow cases. In turbulent flow, the heat transfer coefficient onthe heat source surface is high enough to reduce the heat flow to the substrate to aninsignificant level. Gorski and Plumb (1990, 1992) performed numerical analyses onthe two-dimensional (Fig. 6.20a) and rectangular (Fig. 6.20b) patch problems. Thecross section of the heat source and the substrate is shown in Fig. 6.21.
The fluidflow is described by the analytical solution of Blasius for the laminar boundary layer,while the substrate is assumed to be infinitely thick and the numerical solutions arecorrelated for the two-dimensional strip by 0.71 ksub 0.0570.53 sNu = 0.486Pe(6.174)xskfwhere Nu = h̄s /kf , h̄ is the average heat transfer coefficient based on the heat sourcearea (per unit width normal to the page in Fig.
6.21), s is the heat source length, xs isthe distance between the leading edge of the substrate and that of the heat source, ksubis the substrate thermal conductivity, and kf is the fluid thermal conductivity. Pe is thePéclet number, Pe = U0 xs /α, where U0 is the free stream velocity and α is the fluidthermal diffusivity. Equation (6.174) correlates the numerical solutions within 5% inthe parameter ranges 103 ≤ Pe ≤ 105 , 0.10 ≤ ksub /kf ≤ 10, and 5 ≤ xs /s ≤ 100.For the rectangular patch:0.63 2sP s 0.18ksub0.480.60Pe=1(6.175)c2xs + s2AkfNu =0.70 2sP s 0.07ksub 0.43Pe0.52=10(6.176)c2xs + s2AkfFigure 6.21 Two-dimensional flush-mounted heat source.BOOKCOMP, Inc. — John Wiley & Sons / Page 491 / 2nd Proofs / Heat Transfer Handbook / Bejan[491], (53)Lines: 2425 to 2457———*17.97812pt PgVar———Short PagePgEnds: TEX[491], (53)492123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSwhere Nu is as defined for the two-dimensional strip, Pec = Uo (2xs +s )/2α and A/Pis the source surface area/perimeter ratio.
The correlations of eqs. (6.175) and (6.176)are valid for 103 ≤ Pec ≤ 105 , 5 ≤ (2xs + s ≤)2s ≤ 150, and 0.2 ≤ ws /s ≤ 5,where ws is the heat source width. Additional detailed treatments of the subject havebeen compiled by Ortega (1996).6.6.2 Two-Dimensional Block ArrayFigure 6.22 shows the pertinent dimensions, all nondimensionalized by the plate spacing for the two-dimensional block array. Arvizu and Moffat (1982) performed experiments for heat transfer from aluminum blocks in forced airflow in such channels.The parameter ranges covered are Ph = 1/2, 1/4.6, and 1/7, s = 0.5, 1, 2, 3, 6, and8, and 2200 < RePh = UPh /ν < 12,000 Ph = dimensional block height. For s ≥ 2,the heat transfer coefficients for a fixed Ph almost collapse into a single curve. Theeffect of Ph on heat transfer is large for tightly placed blocks; for s = 0.5, when Phis increased from 1/7 to 1/2, the Nusselt number almost doubles.
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